In order to put your partner on the lender would most likely do a remortgage in joint names.
A transfer would also have to be done to put the property into joint names which would need to be registered at the Land Registry with the new mortgage deed and the relevant Land Registry fee payable.
Any figure in the transfer is subject to SDLT but anything below �125k will not attract SDLT (over �150k if in a disadvantaged area).
No money needs to change hands but when the lender remortgages could you use the equity from his property to reduce the amount of remortgage required and have a deed of trust drawn up as to your respective interests in the property should you split and/or if the property is sold so your relative contributions are protected.
Take into account any financial factors such as early repayment penalties which could have a bearing so you do things in the most economic way possible. You may even find you get a better mortgage deal with another lender with the capital reduction.
Make sure you take appropriate advice as to whether to hold the property as joint tenants (50/50) or tenants in common in whatever shares are decided (ie whether if one of you dies their share goes automatically to the other half or if one of you dies their share is dealt with according to their will/intestacy).
Make sure you have considered all tax implications of having a second rental property.