Unless her earnings are high enough she won't be able to remortgage to remove him from the mortgage unless you have a sum to put in as a gift not a loan (the help from us?). Affordability is key and if she needs help from you then she cant afford it.
If you remortgage jointly with you on the mortgage and it is your second home there may be stamp duty implications.
You may well find that it needs to be sold and half of any equity will be his regardless of who put the deposit in if it wasn't legally protected. Similarly if the property is remortgaged in 1 name then he will be eligible for half of any uplift, that said a remortgage is likely to value lower than a new purchase and so there may be little, if any equity.
Has she been to see her current mortgage provider?
From HMRC
Transfer the outstanding mortgage
Joint owners (this may include unmarried couples who are splitting up) may agree that just one of them will take over ownership of a property they bought together, including any outstanding mortgage.
In this case the person taking ownership will pay SDLT on the total chargeable consideration of the following (either or both), if it exceeds the SDLT threshold:
any cash payment that one of the couple makes to the other for their share
the proportion of the outstanding mortgage that belongs to the share of the property being transferred
Example 3
A couple own a house equally together and:
it’s valued at £400,000
they’ve equity in the property of £300,000
they’ve an outstanding mortgage of £100,000
They transfer ownership so that one of them will have sole ownership of the property. The new sole owner:
pays cash for half of the equity - £150,000
becomes responsible for the other person’s half of the outstanding mortgage - £50,000
The consideration for SDLT is £200,000, made up of the:
cash payment
50% share of the outstanding mortgage
The new sole owner pays £1,500 SDLT (0% of £125,000 + 2% of £75,000) and must tell HMRC by filling in an SDLT return.