Rutherford, 26, will play in all three forms of the game while Dilshan plays in the Caribbean Premier League.
Rutherford, who is part of the Kiwi Test side touring England this summer, has played 16 Tests, scoring 755 runs.
"Hamish is a talented top-order batsman," said elite performance director Graeme Welch.
"He has international pedigree in both the long and shorter formats which will be key as he joins us at what is a busy part of the summer."
Rutherford also had a spell playing for Essex during the 2013 season.

Michelle O'Neill told the BBC she still intends to publish the plan later this month, with an indicative budget.
On Thursday, Mrs O'Neill said there was "no longer an executive in place to agree a budget for an elective-care plan".
But she had talks on Friday with the finance minister about the funding.
The Stormont institutions face collapse following the resignation of Martin McGuinness as deputy first minister.
On Friday, the health minister said she "remains committed" to publishing the strategy to tackle waiting lists which she said were "totally unacceptable" to her.
She added that the amount of money needed was "sizeable" and that she was still in conversations with Mairtin O'Muilleoir looking at all the issues involved.
While the health minister says she will push ahead with publishing the plan, as there is no budget or executive in place, its implementation will be more difficult.

Her death certificate said in January that the 60-year-old, best known for her role as Princess Leia, had suffered a cardiac arrest.
But in a statement the LA coroner said the exact cause was unknown.
Sleep apnoea is a common condition in which a person stops breathing during sleep, either for a few seconds or minutes.
As well as listing sleep apnoea as a cause of death, the coroner's statement cited other factors, including heart disease and drug use.
In a statement released to People magazine after the coroner's ruling, Fisher's daughter Lourd said: "My mom battled drug addiction and mental illness her entire life. She ultimately died of it.
"She was purposefully open in all of her work about the social stigmas surrounding these diseases."
Her manner of death would be listed as undetermined, the coroner's statement said.
Fisher had been on tour promoting her book The Princess Diarist when she was taken ill on a flight from London to Los Angeles on 23 December.
She never regained consciousness and died on 27 December at the Ronald Reagan UCLA Medical Centre.
Her mother, actress Debbie Reynolds, died the following day.
The daughter of singer Eddie Fisher and Debbie Reynolds, Carrie Fisher made her film debut opposite Warren Beatty in 1975's Shampoo.
She also appeared in The Blues Brothers, When Harry Met Sally and Hannah and Her Sisters, and provided the voice of Peter Griffin's boss Angela in the adult animated sitcom Family Guy.
But her enduring fame is thanks to her role as Princess Leia in the original Star Wars trilogy. She will appear in the latest Star Wars sequel, The Last Jedi, which is to be released in December.
Derbyshire have signed New Zealand opening Test batsman Hamish Rutherford as cover for Tillakaratne Dilshan from mid-June until the end of July.
[NEXT_CONCEPT]
The health minister has given further clarification around the publication of a new strategy to tackle hospital waiting times in Northern Ireland.
[NEXT_CONCEPT]
Star Wars actress Carrie Fisher died from sleep apnoea and "other factors", the Los Angeles County coroner says.